In geometry we learn about how the sides of polygons relate to the angles in the polygons, but we have not learned how to calculate an angle if we only know the lengths of the sides. Trigonometry (pronounced: trig-oh-nom-eh-tree) deals with the relationship between the angles and the sides of a right-angled triangle. We will learn about trigonometric functions, which form the basis of trigonometry.
Work in pairs or groups and investigate the history of the foundation of trigonometry. Describe the various stages of development and how the following cultures used trigonometry to improve their lives.
The works of the following people or cultures can be investigated:
Cultures
Ancient Egyptians
Mesopotamians
Ancient Indians of the Indus Valley
People
Lagadha (circa 1350-1200 BC)
Hipparchus (circa 150 BC)
Ptolemy (circa 100)
Aryabhata (circa 499)
Omar Khayyam (1048-1131)
Bhaskara (circa 1150)
Nasir al-Din (13th century)
al-Kashi and Ulugh Beg (14th century)
Bartholemaeus Pitiscus (1595)
You should be familiar with the idea of measuring angles from geometry but have you ever stopped to think why there are 360 degrees in a circle? The reason is purely historical. There are 360 degrees in a circle because the ancient Babylonians had a number system with base 60. A base is the number at which you add another digit when you count. The number system that we use everyday is called the decimal system (the base is 10), but computers use the binary system (the base is 2). 360 = 6 × 60 so for them it made sense to have 360 degrees in a circle.
There are many applications of trigonometry. Of particular value is the technique of triangulation, which is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. GPSs (global positioning systems) would not be possible without trigonometry. Other fields which make use of trigonometry include astronomy (and hence navigation, on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
Select one of the uses of trigonometry from the list given and write a 1-page report describing how trigonometry is used in your chosen field.
If ▵ABC is similar to ▵DEF , then this is written as:
Figure 9.1.
Then, it is possible to deduce ratios between corresponding sides of the two triangles, such as the following:
The most important fact about similar triangles ABC and DEF is that the angle at vertex A is equal to the angle at vertex D, the angle at B is equal to the angle at E, and the angle at C is equal to the angle at F.
In your exercise book, draw three similar triangles of different sizes, but each with 
; 
and 
. Measure angles and lengths very accurately in order to fill in the table below (round answers to one decimal place).
Figure 9.2.
| Dividing lengths of sides (Ratios) | ||
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What observations can you make about the ratios of the sides?
These equal ratios are used to define the trigonometric functions.
Note: In algebra, we often use the letter x for our unknown variable (although we can use any other letter too, such as a , b , k , etc). In trigonometry, we often use the Greek symbol θ for an unknown angle (we also use α , β , γ etc).
We are familiar with a function of the form f(x) where f is the function and x is the argument. Examples are:
The basis of trigonometry are the trigonometric functions. There are three basic trigonometric functions:
sine
cosine
tangent
These are abbreviated to:
sin
cos
tan
These functions are defined from a right-angled triangle, a triangle where one internal angle is 90 ∘ .
Consider a right-angled triangle.
Figure 9.3.
In the right-angled triangle, we refer to the lengths of the three sides according to how they are placed in relation to the angle θ . The side opposite to the right angle is labelled the hypotenus, the side opposite θ is labelled opposite, the side next to θ is labelled adjacent. Note that the choice of non-90 degree internal angle is arbitrary. You can choose either internal angle and then define the adjacent and opposite sides accordingly. However, the hypotenuse remains the same regardless of which internal angle you are referring to.
We define the trigonometric functions, also known as trigonometric identities, as:
These functions relate the lengths of the sides of a right-angled triangle to its interior angles.
One way of remembering the definitions is to use the following mnemonic that is perhaps easier to remember:
| Silly Old Hens |
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| Cackle And Howl |
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| Till Old Age |
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You may also hear people saying Soh Cah Toa. This is just another way to remember the trig functions.
The definitions of opposite, adjacent and hypotenuse are only applicable when you are working with right-angled triangles! Always check to make sure your triangle has a right-angle before you use them, otherwise you will get the wrong answer. We will find ways of using our knowledge of right-angled triangles to deal with the trigonometry of non right-angled triangles in Grade 11.
In each of the following triangles, state whether a , b and c are the hypotenuse, opposite or adjacent sides of the triangle with respect to the marked angle.
Figure 9.4.
Complete each of the following, the first has been done for you
Figure 9.5.
Complete each of the following without a calculator:
Figure 9.6.
Figure 9.7.
For most angles θ , it is very difficult to calculate the values of sinθ , cosθ and tanθ . One usually needs to use a calculator to do so. However, we saw in the above Activity that we could work these values out for some special angles. Some of these angles are listed in the table below, along with the values of the trigonometric functions at these angles. Remember that the lengths of the sides of a right angled triangle must obey Pythagoras' theorum. The square of the hypothenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.
| 0 ∘ | 30 ∘ | 45 ∘ | 60 ∘ | 90 ∘ | 180 ∘ | |
| cos θ | 1 |
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| 0 | – 1 |
| sin θ | 0 |
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| 1 | 0 |
| tan θ | 0 |
| 1 |
| – | 0 |
These values are useful when asked to solve a problem involving trig functions without using a calculator.
Exercise 9.2.1. Finding Lengths (Go to Solution)
Find the length of x in the following triangle.
Figure 9.8.
Exercise 9.2.2. Find